Weekly Euro Zone Forecast, July 7, 2023: Higher Bund Yields, Narrowing Bund Yield Spread

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This week’s 2-year/10-year Bund yield spread is a negative 66.5 basis points, compared to a negative 86.2 basis points last week. As a result, this week’s simulation shows that the probability that the inverted yield curve ends by January 5, 2024 is 15.8%, a change from 1.1% in the prior week.

As explained in Prof. Robert Jarrow’s book cited below, forward rates contain a risk premium above and beyond the market’s expectations for the 3-month forward rate. We document the size of that risk premium in this graph, which shows the zero-coupon yield curve implied by current German Bund prices compared with the annualized compounded yield on 3-month bills that market participants would expect based on the daily movement of government bond yields in 14 countries since 1962. The risk premium, the reward for a long-term investment, is large and remains so over the full maturity range to 30 years. The graph also shows a sharp downward shift in yields in the first few years, as explained below.

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For more on this topic, see the analysis of government bond yields in 14 countries through June 30, 2023, given in the appendix.

In this week’s Euro zone forecast, the focus is on three elements of interest rate behavior: the future probability of the recession-predicting inverted yield curve, the probability of negative rates, and the probability distribution of German Bund yields over the next decade.

We start from the closing German Bund yield curve published daily by the Deutsche Bundesbank and other information sources. Using a maximum smoothness forward rate approach, Friday’s implied forward rate curve shows a quick rise in 1-month rates to an initial peak of 3.77%, compared to 3.65% last week. After the initial rise, there is a decline until rates peak again at 3.27%, versus 3.08% last week, and then decline to a lower plateau at the end of the 30-year horizon.

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Using the methodology outlined in the appendix, we simulate 500,000 future paths for the German Bund yield curve out to thirty years. The next three sections summarize our conclusions from that simulation.

A large number of economists have concluded that a downward sloping yield curve is an important indicator of future recessions. A recent example is this paper by Alex Domash and Lawrence H. Summers. The negative 2-year/10-year German Bund spread is currently at a negative 66.5 basis points.

We measure the probability that the 10-year par coupon Bund yield is lower than the 2-year par coupon Bund for every scenario in each of the first 80 quarterly periods in the simulation.[1] The next graph shows that the probability of an inverted yield remains high, peaking at 84.2%, compared to 98.9% last week, in the 91-day quarterly period ending January 5, 2024.

SAS Institute Inc.

The next graph describes the probability of negative 3-month bill rates for all but the first 12 months of the next 3 decades. The probability of negative rates starts near zero but peaks at 21.9%, versus 20.5% last week, in the period ending December 29, 2028.

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In light of the interest-rate-risk-driven failure of Silicon Valley Bank in the United States on March 10, 2023, we have added a table that applies equally well to banks, institutional investor, and individual investor mismatches from buying long-term German Bunds with borrowed short-term funds. We assume that the sole asset is a 10-year German Bund purchased at time zero at par value of 100 euros. We analyze default risk for four different initial market value of equity to market value of asset ratios: 5%, 10%, 15%, and 20%. For the banking example, we assume that the only class of liabilities is deposits that can be withdrawn at par at any time. In the institutional and retail investor case, we assume that the liability is essentially a borrowing on margin/repurchase agreement with the possibility of margin calls. For all investors, the amount of liabilities (95, 90, 85 or 80) represents a “strike price” on a put option held by the liability holders. Failure occurs via a margin call, bank run, or regulatory take-over (in the banking case) when the value of assets falls below the value of liabilities.

The chart below shows the cumulative 10-year probabilities of failure for each of the 4 possible capital ratios when the asset’s maturity is 10 years. For the 5 percent case, that default probability is 42.16%, compared to 39.24% in the prior week.

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This default probability analysis is updated weekly based on the German Bund yield simulation described in the next section. The calculation process is the same for any portfolio of assets with credit risk included.

In this section, the focus turns to the decade ahead. This week’s simulation shows that the most likely range for the 3-month bill yield in the Bund market in ten years is from 0% to 1%, unchanged from last week. There is a 27.85% probability that the 3-month yield falls in this range, a change from 30.25% one week before. Note the shift downward in the second and third semi-annual periods. For the 10-year Bund yield, the most likely range is from 1% to 2%, also unchanged from last week. The probability of being in this range is 22.82%, compared to 25.25% one week prior.

In a recent post on Seeking Alpha, we pointed out that a forecast of “heads” or “tails” in a coin flip leaves out critical information. What a sophisticated bettor needs to know is that, on average for a fair coin, the probability of heads is 50%. A forecast that the next coin flip will be “heads” is literally worth nothing to investors because the outcome is purely random.

The same is true for interest rates.

In this section we present the detailed probability distribution for both the 3-month bill rate and the 10-year Bund yield 10 years forward using semi-annual time steps. We present the probability of where rates will be at each time step in 1 percent “rate buckets.” The forecast for 3-month bill yields is shown in this graph:

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SASDEU3mBills20230707.xlsx

The probability that the 3-month bill yield will be between 1% and 2% in 2 years is shown in column 4: 29.74%. The probability that the 3-month yield will be negative (as it has been often in Europe and Japan) in 2 years is 8.07% plus 0.92% plus 0.06% plus 0.00% = 9.05% (difference due to rounding). Cells shaded in blue represent positive probabilities of occurring, but the probability has been rounded to the nearest 0.01%.The shading scheme works like this:

The chart below shows the same probabilities for the 10-year Bund yield derived as part of the same simulation.

SAS Institute Inc.

SASDEU10yBund20230707.xlsx

The probabilities are derived using the same methodology that SAS Institute Inc. recommends to its KRIS® and Kamakura Risk Manager® clients. A moderately technical explanation is given later in the appendix, but we summarize it briefly first.

Step 1: We take the closing Bund yield curve as our starting point.

Step 2: We use the number of points on the yield curve that best explains historical yield curve shifts. We note in the following graph that Bund yields span (by rate level and maturity) only 42.06% of the historical experience in 14 countries:

SAS Institute Inc.

For the highest degree of realism in a forward-looking simulation, using the international database is essential. Using daily government bond yield data from 14 countries from 1962 through June 30, 2023, we conclude that 12 “factors” drive almost all movements of government bond yields. The countries on which the analysis is based are Australia, Canada, France, Germany, Italy, Japan, New Zealand. Russia, Singapore, Spain, Sweden, Thailand, the United Kingdom, and the United States of America. No data from Russia is included after January, 2022.

Step 3: We measure the volatility of changes in those factors and how volatility has changed over the same period.

Step 4: Using those measured volatilities, we generate 500,000 random shocks at each time step and derive the resulting yield curve.

Step 5: We “validate” the model to make sure that the simulation EXACTLY prices the starting Bund curve and that it fits history as well as possible. The methodology for doing this is described below.

Step 6: We take all 500,000 simulated yield curves and calculate the probabilities that yields fall in each of the 1% “buckets” displayed in the graph.

We showed in a recent post on Seeking Alpha that, on average, investors have almost always done better by buying long term bonds than by rolling over short term Treasury bills in the United States. That means that market participants have generally (but not always) been accurate in forecasting future inflation and adding a risk premium to that forecast. This study is being updated using the 14-country data set in coming weeks.

Daily government bond yields from the 14 countries listed above form the base historical data for fitting the number of yield curve factors and their volatility. The Bund historical data is provided by the Deutsche Bundesbank. The use of the international bond data increases the number of observations to more than 107,000 and provides a more complete range of experience with both high rates and negative rates than a Bund data set alone provides.

The modeling process was published in a very important paper by David Heath, Robert Jarrow and Andrew Morton in 1992:

Econometrica

For technically inclined readers, we recommend Prof. Jarrow’s book Modeling Fixed Income Securities and Interest Rate Options for those who want to know exactly how the “HJM” model construction works.

The number of factors, 12 for the 14-country model, has been stable for some time.

Footnotes:

[1] After the first 20 years in the simulation, the 10-year yield cannot be derived from the initial 30-year term structure of yields.

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This article was written by

Donald R. van Deventer is a Managing Director in the Center for Applied Quantitative Finance at SAS Institute, Inc. Prior to the acquisition of Kamakura Corporation by SAS on June 24, 2022, Dr. van Deventer was the Chairman and Chief Executive Officer of Kamakura Corporation. He founded the Kamakura Corporation in April, 1990. The second edition of his book, Advanced Financial Risk Management (with Kenji Imai and Mark Mesler) was published in 2013. Dr. van Deventer was senior vice president in the investment banking department of Lehman Brothers (then Shearson Lehman Hutton) from 1987 to 1990. During that time, he was responsible for 27 major client relationships including Sony, Canon, Fujitsu, NTT, Tokyo Electric Power Co., and most of Japan's leading banks. From 1982 to 1987, Dr. van Deventer was the treasurer for First Interstate Bancorp in Los Angeles. In this capacity he was responsible for all bond financing requirements, the company’s commercial paper program, and a multi-billion dollar derivatives hedging program for the company. Dr. van Deventer was a Vice President in the risk management department of Security Pacific National Bank from 1977 to 1982. Dr. van Deventer holds a Ph.D. in Business Economics, a joint degree of the Harvard University Department of Economics and the Harvard Graduate School of Business Administration. He was appointed to the Harvard University Graduate School Alumni Association Council in 1999 and served through 2021. Dr. van Deventer was Chairman of the Council for four years from 2012 to 2016. From 2005 through 2009, he served as one of two appointed directors of the Harvard Alumni Association representing the Graduate School of Arts and Sciences. Dr. van Deventer also holds a degree in mathematics and economics from Occidental College, where he graduated second in his class, summa cum laude, and Phi Beta Kappa. Dr. van Deventer speaks Japanese and English.

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